Abstract. The purpose of this paper is twofold. First, I
want to point out that Wittgenstein's Tractatus contains a
clear and remarkably modern example of a theory of supervenience.
And secondly, I want to argue that this theory of supervenience may
be interpreted as a weak form of a principle of
truth-functionality--which may exactly be the form of this
principle which Wittgenstein himself had in mind.

The purpose of this paper is twofold. First, I want to point out
that Wittgenstein's Tractatus contains a clear and
remarkably modern example of a theory of supervenience. And
secondly, I want to argue that this theory of supervenience may be
interpreted as a weak form of a principle of
truth-functionality--which may exactly be the form of this
principle which Wittgenstein himself had in mind.

"Supervenience" means something like "dependence". When
something depends on another thing, we also say that that thing
supervenes on the other one. Modern supervenience theorists see a
lot of relations of supervenience between various domains. For
example, it is often said that the moral supervenes on the
non-moral. When two organisms are alike in all non-moral respects,
they cannot possibly differ in some moral respect; when someone's
biography is exactly the same as that of the present president of
Austria in all biological, psychological and other non-moral
respects, he or she cannot fail to have the same moral status as
that president has (whichever that may be). No moral difference
without some other difference. It may be the same with the mental:
any exact physical duplicate of me must necessarily have precisely
the same mind as I have. No mental difference without some physical
difference.

The interesting thing about supervenience is that it is a much
weaker notion than reducibility. Moral facts may depend on
non-moral ones, but no one may be able to spell out the dependence
of the supervenient superstructure on the basis in detail; no one
may be able to give a "reduction" in terms of non-moral facts.
Similarly, the mental may fully depend on the physical, but no one
may ever be able to describe the mechanism (or logic?) of the
dependence in detail. This is an advantage of the notion, for
physicalists often want to defend only a vague, general form of
dependence, without wanting to posit any lawlike relationships.

Haugeland has given a definition of supervenience which nicely
fits our purposes:

Two worlds in a class of possible worlds are discernible
with a given language just in case there is a sentence of this
language which is true at one, and not at the other. [...] A
language weakly supervenes on another language (relative to
a class of possible worlds) just in case any two worlds in the
class of possible worlds which are discernible with the former
language are also discernible with the latter. ([1], p. 97.)

On seeing the above definition, any reader of the
Tractatus will immediately be reminded of section 4.26 of
this work:

If all true elementary sentences are given, the result is a
complete description of the world. The world is completely
described by giving all elementary sentences, and adding which of
them are true and which false.

In other words, when you take any "possible world" (as
Wittgenstein calls it: Notebooks 19.9.1916) different from
"the real world", there is always at least one elementary sentence
which is true in only one of both worlds. Wittgenstein does not
state whether he would want to apply this principle to all
possible worlds, but we may safely assume that he did. So let us
extend TLP 4.26 to the thesis that worlds may always be
discerned by elementary sentences. No difference between worlds
without some elementarily describable difference.

Applying Haugeland's definition, it will be clear that we then
have a principle of supervenience here. Everything which may be
discerned with the whole language may already be discerned by means
of the elementary sentences alone, and therefore the whole language
supervenes on its subset of elementary sentences. We may extend the
definition and likewise say that all facts supervene on elementary
facts (Sachverhalte, described by elementary sentences),
that all properties supervene on elementary properties (attributed
by elementary sentences), etc.

Wittgenstein states his thesis of truth-functionality in
Tractatus 5-5.01:

The sentence is a truth-function of the elementary sentences.
(The elementary sentence is a truth-function of itself.) The
elementary sentences are the truth-arguments of sentences.

The principle of truth-functionality as stated here is weaker
and more liberal than the definitions which we have become
accustomed to nowadays. According to present-day definitions, a
sentence cannot be truth-functional unless its truth-value is some
function of the truth-values of the subsentences it contains
and of the way it is built up from these. The truth-values of other
sentences do not matter. However, the Tractatus does not say
that only the truth-values of the elementary subsentences of
a sentence matter as to its truth-value. The latter truth-value may
as well be a function of the truth-values of all elementary
sentences.

Let us, for the moment, interpret the Tractarian principle of
truth-functionality in the latter way. Thus, this principle asserts
that the truth-values of all elementary sentences (not necessarily
only the ones contained in the sentence as subsentences) jointly
determine the truth-values of all sentences. Given what Carnap
called a state-description--a set which, for each elementary
sentence, contains either this sentence or its negation, and no
other elements--any sentence may assume only one
truth-value.

When we accept this weak formulation, the principle of
truth-functionality is easily seen to follow from the supervenience
principle we have just mentioned. For if the thesis of
truth-functionality did not hold, two worlds could verify the same
elementary sentences and yet differ as regards the truth-value of
some other sentence. These worlds would hence not be
completely described by elementary sentences and violate the
principle of supervenience.

The converse implication does, of course, not hold: we may
conceive of a language which is purely truth-functional but unable
to describe any one world completely. Such a language would,
however, not be in accordance with the Tractatus.

Now did Wittgenstein really have such a weak principle of
truth-functionality in mind, or did he accept the principle in its
stronger, full-blown modern version? I think it is hard to find
evidence for the latter view. Wittgenstein never explicitly
banishes modal and doxastic constructions from the ideal language
he had in mind. They seem to be perfectly in order, provided they
are truth-functional; for otherwise the principle of supervenience
of language on elementary language would be violated. And why
should they be prohibited, after all? This would not only lead to a
drastic impoverishment of language (which is nowhere explicitly
advocated in the Tractatus), there is also no justification
for it on syntactic grounds. Modalizing a sentence is no more
mysterious than negating it. And indeed, most of Wittgenstein's
remarks on syntactic operators seem to apply equally well to all
operators, including modal and other ones which we do not longer
call truth-functional nowadays.

Thus, I believe Wittgenstein's principle of truth-functionality,
which would be unduly restrictive otherways, may best be regarded
as being only such a rather weak claim. This also explains why
Wittgenstein himself is so silent about the truth-functionality of,
e.g., intentional ascriptions. They are simply no exception to the
rule. They differ from what we still call truth-functional
compounds nowadays in that their truth-values are not fully
determined by the subsentences they contain and the way they are
built up from these alone; the additional factors that play a role
(according to the Tractatus) are spelled out in [2]. But they are truth-functions of the
elementary sentences nonetheless, so there is no need to pay
special attention to them. It is the same with modal sentences:
their truth-values are the same in all worlds and in all
interpretations which are in accord with the demands the
Tractatus puts on such interpretations, as I explain in [2], and so they are trivially
truth-functional. (Unlike the case of intentional ascriptions,
their truth-values are, however, determined by their
structure alone).

Far from being a drawback, the broad, general nature of the
Tractarian principle of truth-functionality is in fact a point in
its favour. It is an enviable position to be able to claim that
there are some sentences which are basic from an epistemological or
scientific point of view, and which jointly determine all truths
and falsehoods, without being obliged to say how they manage to do
just that. The long history of failed attempts at giving explicit
truth-functional definitions of modal and intentional language may
indeed suggest that it is well-nigh impossible to do the latter,
while the thesis yet remains attractive in its own right. Thus,
Wittgenstein may have been wise in going no further.

Modern theories of supervenience, which also argue for
dependence without committing themselves to reducibility, stem from
the same motivation and share the same appeal. These theories show
that it is possible to be precise and yet not too specific when
giving physicalistic accounts of morality and mind. The
Tractatus shows that the same may be done when one is
formulating sweeping statements on the nature of the semantical
relations between various kinds of sentences.